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## Re: [ontolog-forum] Form and content

 To: "Adrian Walker" "[ontolog-forum]" "sean barker" Sun, 13 Dec 2009 17:38:17 -0000 <7DE5D5D631C04BABB19F3CC7E16C93A0@SMB>
 Adrian       I'm not sure exactly what your function s does, but from what you describe, it just decides whether a string can be mapped to the integers i a farily obvious way. While that gives you a countably infinte set of symbols, that is not the same as the integers.       If I remember correctly, you need sixteen axioms to define the rational numbers, which include axioms such as "One is not equal to zero" (or equivalent) and axioms for ordering numbers. The integers differ from the rationals by not having a multiplicative inverse - and therefore not having the axioms for multaplicative inverse. That is, the set of axioms you choose defines the mathematical structures you generate, be they grupoids, groups, rings, fields etc., and conversely, if you don't use all the axioms for the integers, then the structures you generate include structures which have different properties to those of the integers.       My understanding of Godel's theorem is that once you can do the integers, you necessarily in a system which contains undecidable propositions. This does not prevent you from having systems which are decideable, but the "numbers" they contain will not actually be the integers although they may have some properties which look the same.   Regards   Sean Barker Bristol ----- Original Message ----- From: Adrian Walker To: sean barker Sent: Saturday, December 12, 2009 6:54 PM Subject: Re: [ontolog-forum] Form and content Sean,Yes, if you let the full power of logic loose above a description of the integers, the situation is undecidable.At the other extreme, you can just write simple grammar along lines                   int --> s(0) | s(int)         and you can use it to decide whether any input string is a positive int. **So my point is that a useful Logic (or other description method) including the integers should be designed to be as expressive as possible, while stopping short of undecideability.                         Cheers,  -- Adrian**  And similarly for integers in everyday decimal notationInternet Business LogicA Wiki and SOA Endpoint for Executable Open Vocabulary English over SQL and RDFOnline at www.reengineeringllc.com    Shared use is freeAdrian WalkerReengineering On Sat, Dec 12, 2009 at 12:09 PM, sean barker wrote: Adrian,       I believe the short answer is no - it is a result of one of Godel's theorems. I guess the starting point here is "Some Metamathematical Results on Completeness and Consistency", 1930:   "If to the Peano axioms we add the logic of Principia Mathematica (with the natural numbers as the individuals) together with the axiom of choice (for all types) we obtain a formal system S for which the following theorem's hold: I : The system S is not complete: that is, it contains propositions A for which neither A or NOT A is provable..."   I don't know the area well enough to say at which point non-completenss comes in, whether it is the axioms of Principia Mathematica or the axiom of choice, or even just Peano's axiomatisation.   Regards,   Sean Barker, Bristol ----- Original Message ----- From: Adrian Walker To: sean barker Sent: Thursday, December 10, 2009 9:46 PM Subject: Re: [ontolog-forum] Form and content Hi Sean --You wrote...of course, you wouldn't be able to axiomatise even the integers in many description logics, since they are designed to prevent you describing problems that are undecidable.If that is indeed so, it would seem to be shortcoming of those particular DLs.  There are simple recursive definitions of the integers, and the question of whether an arbitrary string satisfies such a definition is decidable.                      Cheers,  -- AdrianInternet Business LogicA Wiki and SOA Endpoint for Executable Open Vocabulary English over SQL and RDFOnline at www.reengineeringllc.com    Shared use is freeAdrian WalkerReengineering On Thu, Dec 10, 2009 at 3:49 PM, sean barker wrote:   Adrian,       The great thing about numbers is there are so many different sorts to choose from: Naturals, integers, rationals, reals, complex, quarternons and so on, not to mention the modulo groups, the transfinite numbers and the floating point numbers (which, not being a metric space, make a mess of most theorems about numbers). I suspect if you get into the hard maths, then that will raise questions about the axiom of choice and whether you should use it. And, of course, you wouldn't be able to axiomatise even the integers in many description logics, since they are designed to prevent you describing problems that are undecidable.   Sean Barker Bristol, UK   From: ontolog-forum-bounces@xxxxxxxxxxxxxxxx [mailto:ontolog-forum-bounces@xxxxxxxxxxxxxxxx] On Behalf Of Adrian WalkerSent: 10 December 2009 15:20To: [ontolog-forum]Subject: Re: [ontolog-forum] Form and content ``` *** WARNING *** This message has originated outside your organisation, either from an external partner or the Global Internet. Keep this in mind if you answer this message. ``` Hi John,You wrote...anybody who needs a different ontology for numbers can use that instead.Somehow I'm reminded of the saying "the nice thing about standards is that there are so many of them to choose from"Seriously, if we can't agree on a standard for something as basic as a number, what are the chances for interoperability without the need for expensive and continuing human intervention?To get a measure of how far there is still to go on this journey, here's a real world problem description that serves as a nice example of an interoperability requirement:  http://www.economist.com/businessfinance/displaystory.cfm?story_id=15016132                         Cheers,  -- AdrianInternet Business LogicA Wiki and SOA Endpoint for Executable Open Vocabulary English over SQL and RDFOnline at www.reengineeringllc.com    Shared use is freeAdrian WalkerReengineering On Thu, Dec 10, 2009 at 9:23 AM, John F. Sowa wrote: Most people on this list understand the distinction between formand content, but some discussions tend to blur the distinction.I received an offline question related to that point, and I thoughtit might be useful to forward the answer to the list.John Sowa___________________________________________________________________ > May I know what is the difference between semantic network > and ontology?A semantic network is a graphical form for knowledge representation.It can be used to express content of any kind.An ontology is a formal definition of content, which could berepresented in many different KR languages and notations.The distinction between form and content is critical, but someKR languages incorporate some ontology into the basic notation.Therefore, they would combine some content with the form.For example, a temporal logic is likely to have at least a minimalontology for time built into the notation and rules of inference.A KR language that includes arithmetic will have an ontology fornumbers and operations on numbers built into the basic language.Common Logic is a version of logic that is as neutral as possibleabout ontology.  For example, CL includes syntax for numerals, butit does not assume any axioms about the integers represented bythose numerals.  One application might represent integers ofarbitrary length, but another might have an upper limit of 2^63.Some people have complained about the lack of an ontology fornumbers in CL.  But there are many different ontologies that canbe added to CL as needed.  One example is the Mathematical toolkitin the ISO standard for Z notation.  It's a fine ontology, it'sdefined by an ISO standard, and anyone who wants it can use itin conjunction with CL.  But anybody who needs a differentontology for numbers can use that instead.John Sowa
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 Current Thread Re: [ontolog-forum] form and content, (continued) Re: [ontolog-forum] form and content, John F. Sowa Re: [ontolog-forum] form and content, Avril Styrman Re: [ontolog-forum] form and content, John F. Sowa Re: [ontolog-forum] form and content, Patrick Cassidy Re: [ontolog-forum] Form and content, sean barker Message not available Re: [ontolog-forum] Form and content, sean barker Re: [ontolog-forum] Form and content, Adrian Walker Re: [ontolog-forum] Form and content, John F. Sowa Re: [ontolog-forum] Form and content, Adrian Walker Re: [ontolog-forum] Form and content, John F. Sowa Re: [ontolog-forum] Form and content, sean barker <= [ontolog-forum] form and content, FERENC KOVACS [ontolog-forum] form and content, FERENC KOVACS [ontolog-forum] form and content, FERENC KOVACS [ontolog-forum] Form and content, FERENC KOVACS Re: [ontolog-forum] Form and content, Rich Cooper